question in statistics and probabilities

Question

The time between failures of a machine has an exponential distribution with parameter λ . Suppose that the prior distribution for λ is exponential with mean 100 hours. Two machines are observed, and the average time between failures is x = 1223 hours.

(a) Find the Bayes estimate for λ . [Round your answer to five decimal places.]

(b) What proportion of the machines do you think will fail before 1000 hours? [Round your answer to three decimal places.]

Answer 1

Please define failure from a statistical perspective and clarify how there can be a statistical calculation involving a parameter, which has limits and subsets, and an exponential, which is forever expanding?